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Apéry extensions

Vasily Golyshev, Matt Kerr, Tokio Sasaki

Abstract

The Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau-Ginzburg models -- and thus, in particular, as periods. We also construct an ``Apéry motive'', whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard-Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG-models mirror to several Fano threefolds. By describing the ``elementary'' Apéry numbers in terms of regulators of higher cycles (i.e., algebraic $K$-theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG-models are modular families of $K3$ surfaces, and the distinction between multiples of $ζ(2)$ and $ζ(3)$ (or $(2π\mathbf{i})^3$) translates ultimately into one between algebraic $K_1$ and $K_3$ of the family.

Apéry extensions

Abstract

The Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau-Ginzburg models -- and thus, in particular, as periods. We also construct an ``Apéry motive'', whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard-Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG-models mirror to several Fano threefolds. By describing the ``elementary'' Apéry numbers in terms of regulators of higher cycles (i.e., algebraic -theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG-models are modular families of surfaces, and the distinction between multiples of and (or ) translates ultimately into one between algebraic and of the family.

Paper Structure

This paper contains 21 sections, 12 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Generic Laurent polynomials
  3. GKZ system
  4. Periods and residues
  5. Mixed Hodge theory of GKZ
  6. Special Laurent polynomials
  7. Landau-Ginzburg models
  8. Variation of Hodge structure
  9. Quantum $\mathcal{D}$-module
  10. Higher normal functions
  11. Variation of mixed Hodge structure
  12. Admissible and geometric normal functions
  13. $\mathcal{V}_{\phi}$ as a (mixed) higher normal function
  14. $\mathcal{V}_{\phi}$ at infinity
  15. The Problem and some Fano threefold examples
  16. ...and 6 more sections

Key Result

Theorem 1.1

The Apéry numbers of the five Mukai Fano threefoldsThese are, by definition, the rank-one Fano 3-folds arising as complete intersections in the Grassmannians of simple Lie groups other than projective spaces Gv; they are $V_{10},\,V_{12},\,V_{14},\,V_{16}$, and $V_{18}$. are limits of higher normal

Theorems & Definitions (54)

  • Theorem 1.1
  • Proposition 2.1
  • proof : Check:
  • Example 2.2
  • Example 2.3
  • Theorem 2.4: HLYZ
  • Theorem 2.5: Conjectured by KKP, proved by Ha ($n=3$) and Sa
  • Theorem 2.6: HLY
  • Conjecture 2.7: Hyperplane Conjecture HLYLZ
  • Example 2.8
  • ...and 44 more